2 BERNHARD AND SECKMEYER irradiance in the UV is a demanding task [McKenzie et al., 1994]. Diculties arise mainly from the steep decline of the solar

Transcription

1 To appear in the Journal of Geophysical Research, 1999 Uncertainty of measurements of spectral solar UV irradiance G. Bernhard 1 and G. Seckmeyer Fraunhofer Institute for Atmospheric Environmental Research, Garmisch-Partenkirchen, Germany Abstract. Most investigations on the nature and eects of solar ultraviolet (UV) radiation at the Earth's surface require measurements of high accuracy combined with well-dened procedures to assess their quality. Here we present a general evaluation of all relevant errors and uncertainties associated with measurements of spectral global irradiance in the UV. The uncertainties are quantied in terms of dependence of the characteristics of the spectroradiometer, the uncertainty of calibration standards, the solar zenith angle, and atmospheric conditions. The methodologies and equations presented can be applied to most spectroradiometers currently employed for UV research. The sources of error addressed include radiometric calibration, cosine error, spectral resolution, wavelength misalignment, stability, noise, stray light, and timing errors. The practical application of the method is demonstrated by setting up a complete uncertainty table for the mobile spectroradiometer of the Fraunhofer Institute for Atmospheric Environmental Research (IFU). This instrument has successfully participated in several international intercomparisons of UV spectroradiometers. The expanded uncertainty (coverage factor k = 2)for measurements of global spectral irradiance conducted with this instrument varies between 6.3% in the UVA and 12.7% at 300 nm and 60 solar zenith angle. The expanded uncertainties in erythemally and DNA weighted irradiances are 6.1% and 6.6%, respectively. These expanded uncertainties are comparable to uncertainties at the 2 level in conventional statistics. A substantial reduction of these uncertainties would require smaller uncertainties in the irradiance standards used to calibrate the instrument. Though uncertainties caused by wavelength misalignment and noise become prominent in the shortwave UVB, which is the most important spectral range for UV trend detection, the results indicate that the accuracy of the IFU radiometer is sucient to detect long-term trends in UV arising from a 3% change in atmospheric ozone. The detection of trends caused by a 1% change in ozone may bebeyond the capabilities of current instrumentation. 1. Introduction Solar UV radiation can be detrimental to terrestrial and aquatic ecosystems as well as human health. Because of the decline in stratospheric ozone concentrations observed at high and middle latitudes [Harris et al., 1994], it is expected that UV levels at the Earth's surface will increase. If current and future risks from solar UV radiation are to be quantied, measurements of solar spectral UV irradiance of known quality are an 1 Now at Biospherical Instruments Inc., San Diego, California. important prerequisite. This paper focuses on the determination of the uncertainty table for measurements of global spectral UV irradiance E G (), i.e., the radiant energy dq arriving per time interval dt, per wavelength interval d, and per area da on a horizontal surface from all parts of the sky above the horizontal, including the disc of the Sun itself: E G () = dq dt da d : (1) E G () is the most commonly used spectral quantity for UV research. However, the measurement of spectral 1

2 2 BERNHARD AND SECKMEYER irradiance in the UV is a demanding task [McKenzie et al., 1994]. Diculties arise mainly from the steep decline of the solar spectrum in the UVB range, which is due to absorption by atmospheric ozone. The required accuracy of measurements of E G () depends on the intended use of the data. Such uses include, for example, eect studies on biological organisms, providing information to the public about actual UV levels, and investigations on the transfer of radiation through the atmosphere and trend detection. For the last point, in particular, high accuracy is required because trends in UV are expected to be small. The overall uncertainty of a UV measurement depends on the specications of the instrumentation employed, the uncertainty in the calibration sources, and atmospheric conditions. All points will be addressed thoroughly in this paper. Though very important, the inuence of the technical skills of the personnel operating the instruments will not be discussed. All uncertainties reported are consequently based on the assumption that instruments are carefully maintained and characterized. This includes the application of a quality control (QC) plan. In addition, instruments should participate in intercomparison campaigns to assure their quality and to detect systematic errors that would otherwise go undetected. Several such campaigns have been organized in the past in the framework of national, European, and international research programs [e.g., Gardiner and Kirsch, 1995; Webb, 1997; Seckmeyer et al., 1998; Thompson et al., 1997; Kjeldstad et al., 1997]. These publications also include valuable information on the characterization of instruments and on methods to reduce uncertainties by means of both laboratory investigations and the application of correction procedures. Several studies in the past have already addressed the problem of how to estimate the uncertainty of spectral UV measurements. Most of these examinations are related to a specic instrument or to radiometry in general. In contrast, this paper (1) is applicable to a great variety of instruments currently deployed worldwide, (2) focuses on the special needs of solar radiometry in the UV range, (3) includes an analysis about the inuence of dierent atmospheric conditions on the determined uncertainties, and (4) gives examples of uncertainties in biologically weighted irradiance derived from spectral measurements. The results are applicable to scanning spectroradiometers that have a spectral bandwidth between 0.3 and 5.0 nm, have awavelength uncertainty of less than 0.5 nm, use entrance optics with cosine weighting, and are calibrated with tungsten halogen lamps. Most of the spectroradiometers currently used for UV research meet these specications. It is worthwhile to give some examples of investigations conducted to estimate the uncertainty of UV measurements. Bener [1960] gives a detailed uncertainty report for measurements of a spectroradiometer deployed in the 1950s and 1960s in Davos-Platz, Switzerland. However, since his measurements were carried out at 1590 m above sea level and his equipment was quite dierent from current instrumentation, his results can only partly be transferred to today's needs. The instruments that are currently most often used to measure spectral UV irradiance are Brewer ozone spectrophotometers [Brewer, 1973; Bais et al., 1996]. A comprehensive characterization and determination of uncertainties are given by Kohler et al. [1995]. McKenzie et al. [1992] describe the characterization and evaluation of the errors of a spectroradiometer developed in New Zealand. The above studies are only applicable to the particular instrument under test. In contrast, Bais [1997] gives a qualitativeoverview of the errors and uncertainties involved in the measurement ofe G (). In the framework of activities of the Scientic Advisory Group (SAG) on UV measurements established by the World Meteorological Organisation (WMO), a document regarding guidelines for site quality control of UV monitoring was published [Webb et al., 1998]. A standard method of estimating uncertainty is proposed, with special emphasis on comparability between dierent measurement sites. According to Webb et al. [1998], the actual uncertainty of a single measurement at a given site may deviate from this standard estimate. In contrast, our investigation focuses on the actual uncertainty of individual instruments. Recently, a comprehensive textbook on radiometry was published by Kostkowski [1997]. As an example, the measurement of direct solar spectral irradiance between 295 and 315 nm is described. Some errors in spectroradiometry are addressed more comprehensively there than can be done here. However, in contrast to Kostkowski [1997], we focus on the measurement of global rather than direct irradiance and we also consider the needs of UV eect research and the inuence of the atmosphere on uncertainties. In section 3, all error sources and uncertainties arising in solar UV spectroradiometry are described and quantied in a general way. In section 4, the method is then applied to measurements of the \mobile spectroradiometer" of the Fraunhofer Institute for Atmospheric Environmental Research (IFU), establishing an uncertainty report for this instrument. 2. Material and Methods 2.1. Method to Express Systematic Errors and Uncertainties In spectoradiometry, the simple equation E M () = S M() r() (2)

3 SOLAR UV MEASUREMENT UNCERTAINTY 3 is frequently used to determine the spectral irradiance E() produced by the radiation source to be measured. In (2), S M () is the signal of the radiometer when measuring the source, r() is the spectral responsivity of the instrument, and E M () is the measurement result. In practice, the measured irradiance E M () will deviate from the \true" spectral irradiance E() owing to systematic and statistical errors. Systematic errors describe the deviation between the \true value" of the quantity to be measured and the average of a large number of measurements of this quantity carried out under repeatable conditions. In contrast, statistical errors express the uctuation of individual measurements around the average Systematic errors. The deviation of the measured irradiance E M () from the true irradiance E() due to systematic errors can be expressed by E M () = E() R(; x 1 ;x 2 ; :::x n ) (3) where the factor R(; x 1 ;x 2 ; :::x n ) takes into account all sources of systematic error aecting the measurement. If the dierent error sources are independent from each other, which is the case for all errors introduced below unless stated otherwise, R(; x 1 ;x 2 ; :::x n ) can be split into several parts: R(; x 1 ;x 2 ; :::x n ) = R X1 (; x 1 ) R X2 (; x 2 ) ::: R Xn (; x n ): (4) The factors R Xi (; x i ) with (1 i n) represent n dierent sources of systematic error X 1, X 2,..., X n and are dened by R Xi (; x i ) = E X i (; x i ) E() (5) where, again, E() is the true spectral irradiance and E Xi (; x i ) is the idealized irradiance that would be measured if only the error source X i existed. The variable x i expresses the parameter that characterizes the source of error X i. For example, the error source \wavelength shift" explained in section 3.4 and abbreviated by the character W is characterized by the magnitude of the shift. Thus R W (; ) = E W (; )=E() is the factor by which the measured irradiance E M () deviates from E() owing to a wavelength shift of, neglecting all other systematic errors. If a systematic error X i can be completely characterized, the measured irradiance E M () can be corrected for this error simply by multiplying E M () with the correction factor K Xi (; x i )=1=R Xi (; x i ). In reality, systematic errors can never be completely determined and corrected for, and uncertainties in the measurement result are therefore unavoidable Uncertainties. Throughout this paper, the way of expressing uncertainties is in accordance with the International Standards Organization (ISO) [1993]. Here uncertainties evaluated by a statistical analysis of a series of observations are denoted \type A standard uncertainites" u A. Usually, u A is equal to the sample standard deviation of the mean s u A = s = 1 p m s (6) where s is the sample standard deviation derived from m independent measurements of the given quantity. Type A uncertainties can consequently be reduced by increasing the number of measurements. ISO [1993] denotes uncertainties originating from (incompletely corrected) systematic errors as \type B standard uncertainties." These are determined by means other than by the statistical analysis of a series of measurements. According to ISO [1993], the type B standard uncertainty u B related to a systematic error is u B = 1 2 p 3 (a +, a, ) (7) where a + is the estimated upper limit and a, is the estimated lower limit for the quantity being measured. (This denition is based on the assumption that the probability distribution for a measurement result is rectangular; that is, the probability that the true value lies within the interval a + and a, is constant and is zero outside this interval. This is the usual assumption if no information about the actual distribution is available. For details, see ISO [1993] and Kostkowski [1997]). Type B uncertainties cannot be reduced by increasing the number of measurements. Following the recommendations of ISO [1993], all standard uncertainties of type A, u Aj, and type B, u Bi, are combined in quadrature: u comb = s X j u 2 A j + X i u 2 B i (8) Note that (8) is valid only if all uncertainty components are uncorrelated. If this is not the case, a covariance term has to be added to (8); see ISO [1993]. The resulting combined uncertainty u comb is nally multiplied by a coverage factor k to derive the expanded uncertainty U of the nal measurement result. In this paper, the coverage factor is set as k = 2. Thus expanded uncertainties reported herein are comparable to uncertainties on the \2 level" in conventional statistics Radiative Transfer Model The studies concerning the dependence of uncertainties on atmospheric conditions are based on model calculations. The model employed is the pseudospherical version of the radiative transfer model UVSPEC [Dahlback and Stamnes, 1991; Mayer et al., 1997], which has

4 4 BERNHARD AND SECKMEYER been shown to agree well with measurements: systematic dierences between measurement and model were found to range between,11% and +2% for wavelengths between 295 and 400 nm and solar zenith angles (SZA) up to 80 [Mayer et al., 1997]. The model proved to be accurate enough to be used in this study to determine the eects of solar elevation (position of the Sun in degrees measured from the horizon), total ozone, albedo, and aerosol optical depth on spectral UV. 3. Sources of Systematic Errors and Uncertainties In the following, the dierent sources of errors in solar radiometry are compiled in the order of decreasing importance Radiometric Calibration With the radiometric calibration, the spectral responsivity r() of a radiometer is determined r() = S L() E L () : (9) E L () is the irradiance produced by a calibration source, and S L () is the signal of the radiometer when measuring the source. According to (2), the ratio S M ()=r() is then the measured irradiance E M (). In solar UV radiometry, tungsten halogen lamps are usually used as the calibration source. Reference standards based on these lamps are available from national standards laboratories such as the National Institute of Standards and Technology (NIST) in the United States, the Physikalisch Technische Bundesanstalt (PTB) in Germany, or the National Physical Laboratory (NPL) in Great Britain. Further uncertainties in the determination of the instrument responsivity arise from the aging of the standards between their calibration and use, the conditions during lamp operation, and the characteristics of the spectroradiometer Uncertainties in the certicates of reference standards. Table 1 gives an overview of uncertainties in reference standards reported by dierent standards laboratories. In an intercomparison of spectral irradiance standards organized between 1987 and 1990 by the Consultative Committee on Photometry and Radiometry (CCPR), the irradiance scales of 12 national standards laboratories were compared [Walker et al., 1991]. The double standard deviation (2) ofthe measurements of all groups was about 4% in the UV. This equals or exceeds the expanded uncertainties given in Table 1. In order to investigate the dierences of standard lamps, we have compared seven reference standards of type FEL calibrated by PTB. For none of these lamps did the burn time exceed 10 hours after calibration at PTB. Three of these lamps were fabricated Table 1. Expanded Uncertainties (Coverage Factor k = 2) of Reference Standards From Dierent Standards Laboratories According to Calibration Certicates Wavelength, Uncertainty, nm % National Institute of Standards and Technology, United States Lamp Type: FEL 1000 W, T6, Osram-Sylvania Physikalisch Technische Bundesanstalt, Germany Lamp Type: FEL 1000 W, General Electric Physikalisch Technische Bundesanstalt, Germany Lamp Type: FEL 1000 W, T6, Osram-Sylvania National Physical Laboratory, England Lamp Type: M28, 100 W by General Electric, and their calibration refers to the old pyrometer-based spectral irradiance scale of PTB, which was used until 1995 [Sperfeld et al., 1996]. Four lamps were of the newer T6 type made by Osram- Sylvania. Their calibration refers to the new detectorbased PTB scale [Sperfeld et al., 1996]. All lamps were compared in the IFU calibration laboratory using the IFU spectroradiometer. For details, see Bernhard [1997]. For wavelengths between 280 and 700 nm, we found the calibration of the new T6 type lamps to be in agreement with each other, to within 0:3%. This is within the uncertainties given by PTB; see Table 1. In the wavelength range nm, the irradiance scales of the old and new types were found to agree on the 0:5% level. However, at 325 nm, both types disagreed with each other by 4%, which is at the very limit of PTB's uncertainty estimate. A further test with an M28 type lamp calibrated by NPL showed a discrepancy of more than 6% at 300 nm compared with the new PTB scale. These deviations indicate that the uncertainties of lamps disseminated by standards labora-

5 SOLAR UV MEASUREMENT UNCERTAINTY 5 tories may be larger than specied. From our ndings we conclude that 3:5% is a reasonable estimate for the expanded standard uncertainty (k = 2) of current reference standards in the UV. However, this estimate is itself subject to uncertainties because our survey is based on nine lamps only Aging of standard lamps. The irradiance produced by standard lamps changes with burn time. According to PTB, the drift of FEL lamps from General Electric is 0.03%/hour in the visible range and 0.05%/hour at 300 nm [Sperling et al., 1996]. Moreover, sudden changes (jumps) of up to 1% in the radiation output of these lamps may occur at unpredictable intervals. NIST seasons FEL lamps for 40 hours and only lamps that then exhibit a change at nm of less than 0.5% in 24 hours (= 0:02%/h) after this burnin period are used for reference standards [Walker et al., 1987]. PTB reports that the continuous drift of the new FEL lamps from Osram-Sylvania is less than 0:01%/hour at 300 nm [Sperling et al., 1996]. However, we have seen jumps of about 0:5% with these new Osram-Sylvania FEL lamps. For both FEL lamp types, it is therefore advisable to have at least three standards at place that are regularly compared. In addition to the deterioration of FEL standards, we have studied the drift of 100 W lamps of type M28 and of type Halostar from Osram, which are used at IFU as working standards. In the visible the deterioration of ve lamps was measured with a luxmeter of PRC Krochmann by continously monitoring their illuminance over a period of up to 160 hours. The drift in the UV was determined with the IFU spectroradiometer and control lamps, which were only shortly in use for comparison with the test lamps. After a burn-in time of about 10 hours, the lamp irradiance usually decreases by about 0:02%/hour in the visible and by about 0:04%/hour to 0:08%/hour at 300 nm. Although these drifts are somewhat larger than the drifts of FEL lamps, the lamps deteriorate in a very predictable (and thus correctable) manner and exhibit no jumps Operation of standard lamps. The lamp current is the most critical parameter. According to Webb et al. [1994] and Sperling et al. [1996], a 1% change in lamp current leads to a 10% change in E L () at 300 nm. More generally, Kostkowski [1997] states that the percentage change in the spectral irradiance of FEL lamps is approximately 5 (600=) times the percentage change in current, where is the wavelength in nanometers. To check whether this relation is also applicable for our 100 W lamps, we operated a lamp at 8.5 A (nominal current) and 8.4 A (1.2% below nominal current). The dierence in the irradiance was 10% at 300 nm and 5.5% at 600 nm, which agrees well with the formula proposed by Kostkowski [1997]. Intercomparison campaigns have uncovered signicant systematic errors in solar UV measurements due to inaccurate ammeters [Gardiner and Kirsch, 1995]. An uncertainty in lamp current of less than 0:01% is desirable, and this can only be achieved with a high-accuracy current source. The second most important parameter is the distance between the lamp and the reference plane of the radiometer's entrance optics, measured along the optical axis (x axis). NIST certicates specify irradiance values at 500 mm. At this distance the lamp can be regarded as a point source and therefore E L () changes with distance according to the inverse square law: a 1 mm error in distance results in a 0.4% error in irradiance. For PTB standards, which are calibrated at 700 mm, the respective error is reduced slightly to 0.3%. Sperling et al. [1996] investigated the eect of further alignment errors for FEL lamps. For positioning errors of less than 1 mm in the y and z directions (= symmetry axis of the lamp), rotations of less than 0.1 around the y and z axis, and rotations of less than 2 around the x axis, the standard uncertainty inthe irradiance is 0.1%. This accuracy can be achieved only by laser alignment (for details, see Sperling et al. [1996] and Kostkowski [1997]). Alignment errors concerning the entrance optics lead to an additional irradiance uncertainty of 0.1%. Another source of uncertainty is stray light. This is radiation that is reected or scattered from the walls of the calibration room or from the table. Stray light must be suppressed as far as possible, e.g., by baes and black cloths. Methods to determine uncertainties due to stray light are described by Kostkowski [1997]. Further sources of error resulting from the method of lamp operation (e.g., the temperature in the laboratory, vibration, ventilation, change of polarity of the lamp, and a too short lamp warm-up time) are addressed by Webb et al. [1994], Sperling et al. [1996], and Kostkowski [1997] Inuence of the radiometer on the irradiance calibration. The geometry of the radiometer's entrance optics may lead to several further calibration uncertainties. For example, if a translucent quartz plate is used to protect the optics, the optical path of the incident radiation is reduced by refraction: a quartz plate (refraction index 1.5) with a thickness of 2.5 mm reduces the optical path by 2:5 mm, (2:5 mm=1:5) = 0:8 mm, which in turn introduces an irradiance error of 0.3% for a calibration distance of 50 cm. The error value may change if a dome rather than a plate is used for protection, because the spherical geometry of the dome prevents a rectilinear propagation of the incident radiation and multiple reections between diuser and dome may occur. A method to calculate the reference plane for nonplanar diusers is described by Bernhard and Seckmeyer [1997]. The irradiance produced by a calibration lamp refers to the average irradiance on a receiver with an area

6 6 BERNHARD AND SECKMEYER specied in the lamp's certicate. For example, PTB species an area of 20 mm 10 mm. Since the radiation eld of a FEL lamp is not perfectly isotropic [Sperling et al., 1996], further errors occur if the actual area of the entrance optics deviates from that specied. On the basis of Sperling et al. [1996], we have estimated this standard uncertainty to be about 0.1% for entrance optics smaller than 20 mm 10 mm. The degree of polarization for FEL lamps is about 3% [Kostkowski, 1997]. This introduces a further uncertainty ifentrance optics do not depolarize completely Interpolation of calibration values. Standards laboratories only provide calibration points in steps of 5 to 20 nm. Because of the curvature in the lamp's spectrum, which is similar to the spectrum of blackbody radiation, linear interpolation may lead to errors of up 4%. NIST recommends the following equation to calculate irradiance values between the calibration points given in a certicate [Walker et al., 1987]: E L () =[A 0 +A 1 +:::+A n n ],5 exp(a+b=); (10) where a, b, A 0, A 1,... A n are t parameters. Spectral irradiance values predicted using (10) have an uncertainty of 0.5%. We prefer an interpolation based on cubic splines. In detail, the logarithm of the given calibration points is calculated, the result is interpolated to the required wavelength steps using natural cubic splines, and nally the antilogarithm is applied. Thus the interpolation is based on values ranging within 1 order of magnitude, which leads to smaller interpolation error than the interpolation without forming the logarithm. To estimate the standard uncertainty of this method, we started with a blackbody function similar to an actual lamp spectrum, selected points at the same wavelengths as given in certicates of PTB (i.e., between 250 and 300 nm points are given in 10 nm steps and in 20 nm steps afterward), performed the interpolation, and nally compared the result with the original blackbody function. Except for the region between the rst and second calibration point (i.e., the range between 250 and 260 nm, which is irrelevant for solar measurements), original and interpolated values lay within 0:1%. This uncertainty estimate is only valid, however, if lamps have no absorption or emission lines, which is not necessarily true for FEL lamps [Kostkowski, 1997] Cosine Error The signal of a radiometer for measuring irradiance should be proportional to the cosine of the angle # between the direction of the incident radiation and the normal of the radiometer's entrance optics. The deviation from this ideal response is called cosine error. The ratio R C () of the irradiance measured with a radiometer, which is aected by the cosine error, to the true irradiance is R C () = Z Z (2) (2) L(; #; ') C(; #; ') sin(#) d# d' L(; #; ') cos(#) sin(#) d# d' (11) where ' is the azimuth angle of the incident radiation, L(; '; #) is the spectral radiance at the radiometer's entrance optics, and C(; #; ') is the angular response of the entrance optics, normalized to 1 at # =0. Ideally, C(; #; ') should be equal to cos(#). When global spectral irradiance E G () is the measurand, (11) can be split in two parts, separating the contributions from the direct Sun and the diuse (i.e., scattered) sky radiation [Seckmeyer and Bernhard, 1993; Grobner et al., 1996; Bais et al., 1998]: R C () = q() C(; # Sun;' Sun ) + [1, q()] D(); cos(# Sun ) (12) where the angles # Sun and ' Sun are solar zenith and azimuth angle, respectively, and q() is the ratio between direct irradiance E D () and global irradiance E G (). (E D () is the irradiance produced by unscattered radiation on a horizontal surface. The dierence between global and direct irradiance is denoted sky irradiance E S (): E S () =E G (), E D ()). D() is the ratio of the sky irradiance measured with a radiometer that has a cosine error to the true sky irradiance E S (). D() is calculated with (11), where the integration range excludes the solid angle of the Sun. UV measurements can be corrected for the cosine error by multiplying the measurement result with the correction factor K C () = 1=R C (). Because of the uncertainties in q(), C(; # Sun ;' Sun ), and D(), the accuracy of the correction is limited. The standard uncertainty u RC () ofr C () is expressed by u RC () u q 2 u C = C(; #Sun ;' Sun ) + cos(# Sun ) 2 q() u cos(# Sun ) 2 C +[1, q()] 2 u 2 D 0:5 ; 2 u D, D() 2 u 2 q ; 2 i 0:5 (13) where u q, u C and u D are the standard uncertainties of q(), C(; # Sun ;' Sun ), and D(), respectively. The magnitudes of these uncertainties depend very much on the instrument specications and the prevailing atmospheric conditions, as discussed below Range and uncertainty of C(; #; '). Many UV spectroradiometers currently deployed ex-

7 SOLAR UV MEASUREMENT UNCERTAINTY 7 hibit deviations from the ideal cosine of more than 10% at # = 60. For larger # the deviation may be even greater [e.g., Gardiner and Kirsch, 1995; Webb et al., 1994; Webb, 1997; Seckmeyer et al., 1998]. Therefore the cosine error is one of the most important sources of error in solar spectroradiometry. However, entrance optics with smaller cosine errors are now available [Bernhard and Seckmeyer, 1997; Bais, 1998]. The cosine error may also depend signicantly on wavelength and azimuth angle and, additionally, can change with time. For example, instruments that have an integrating sphere may have a pronounced azimuthal dependence [McKenzie et al., 1993; Webb, 1997] and the coating of the sphere may age. This complicates the application of correction algorithms. C(; #; ') may also depend on the polarization of the incident radiation. Since sky radiance and the radiation of an FEL lamp are polarized, further uncertainties have to be considered. To estimate their magnitude, measurements of the radiance distribution of sky radiation on dependence of polarization are required. Although this is possible [Coulson, 1988; Liu and Voss, 1997; Voss and Liu, 1997], these measurements are rather elaborate and are probably too time-consuming to be performed within a spectrum of global irradiance. A more satisfactory solution is to choose entrance optics with low dependence on polarization. The usual method to determine C(; #; ') is to mount the radiometer on a turn table with its axis going through the center of the reference plane of the radiometer's entrance optics. The radiometer then measures the irradiance of a xed lamp as a function of #, ', and. To reduce uncertainties in the determination of C(; #; '), the integration time of the radiometer must be suciently long. For details, see Seckmeyer and Bernhard [1993], Kostkowski [1997], Feister et al. [1997], and Webb et al. [1994]. Uncertainties in the determination of C(; #; ') arise mainly from misalignment. For example, if the zero position of the turntable is wrong by 1, C(; #; ') is in error by 3%at# =60 and by about 20% at # =85. The standard uncertainty of the zero position of our facility to measure the angular response of entrance optics is 0.08 ; the uncertainty in positioning the center of the entrance optics above the resolving axis is 0.8 mm in both relevant directions. These uncertainties in alignment lead to a standard uncertainty (k = 1)in C(; #; ') of 1.1% at # =60 and 2.8% at Range and uncertainty ofq(). The ratio q() =E D ()=E G () depends on all the parameters that aect UV radiation at the Earth's surface. The ideal method to determine q() is to measure E D () and E G () simultaneously with two independent spectroradiometers. If E D () and E G () are measured alternately with one instrument, a time-interpolation is necessary, and changing atmospheric conditions (e.g., moving clouds) lead to higher uncertainties compared to the simultaneous schedule. The ratio q() can also be approximated by a radiative transfer model, as suggested by McKenzie et al. [1992] and Seckmeyer and Bernhard [1993]. We have recalculated this ratio with the more accurate UVSPEC model introduced in section 2.2 for a variety of SZAs and three dierent aerosol optical depths (). The results are shown in Figure 1. Further input parameters of the model were: total ozone 320 Dobson units (DU), Angstrom's turbidity parameter = 1, surface albedo = 0.05, altitude = 730 m. Clearly, the ratios depend q( ) q( ) q( ) = 0.0 = 0.5 = 1.0 Sun = Wavelength [nm] Figure 1. Ratio q() between direct and global spectral irradiance in dependence of wavelengths for different zenith angles # Sun and aerosol optical depths (). The values are based on UVSPEC model calculations assuming a total ozone column of 320 Dobson units (DU). Aerosol optical depths at 300 nm are (top) (300 nm) = 0:0; (middle) (300 nm) = 0:5; and (bottom) (300 nm) = 1:0. on solar zenith angle and aerosol properties, and these parameters are therefore necessary for a meaningful estimate of q(). Fortunately, q() depends only slightly on total ozone [Zeng et al., 1994]. 0

8 8 BERNHARD AND SECKMEYER For overcast conditions, when the Sun is always hidden, q() is eectively zero resulting in R C () =D(). For broken cloud situations the application of current one-dimensional models is not appropriate and simultaneous measurements of E D () and E G () may be necessary. Feister et al. [1997] use global and diuse irradiance measurements of two broadband UVB sensors to determine q() and, based on that, correct measurements of a Brewer spectrophotometer for the cosine error. The advantage of this method is that diuse and global data are available simultaneously without the need of operating two expensive spectroradiometers. Feister et al. [1997] include a comprehensive uncertainty estimate of the method and a parameterization of q() as a function of, # Sun, and (350 nm) Range and uncertainty of D(). To estimate the diuse ratio D(), it is often assumed that the sky radiance L(; #; ') is isotropic; that is, L(; #; ') does not depend on # and '. In this case, D() simplies to D isotr () = 2 Z =2 0 C(; #; ') sin(#) d#: (14) Grobner et al. [1996] have investigated the deviation of D isotr () from the true factor D() for cloudless skies by measuring sky radiance L(; #; ') in addition to their global measurements of E G (). The study takes into account cloudless sky situations for varying atmospheric and geographic conditions (SZA between 10 and 80 and aerosol optical depths at 320 nm between 0.03 and 0.6). For their instrument, D isotr () is 0.883; that is, irradiance produced by isotropic sky radiance is underestimated by 11.7%. The true factor D() derived from the radiance measurements is 0:833 0:03 at 310 nm, 0:880 0:03 at 320 nm, 0:870 0:035 at 350 nm, and 0:86 0:05 at 400 nm. The plus/minus ranges of D() are maximum variations caused by the dependence of D() on the SZA and atmospheric conditions. The deviation of D() from D isotr () increases with the inhomogeneity of sky radiance. In the study of Grobner et al. [1996] the inhomogeneity was highest at a high-altitude station and increased with increasing wavelength. All the values given above are valid only for the entrance optics used by Grobner et al. [1996]. For a different instrument we propose to use its diuse ratio D isotr () for all atmospheric conditions, if no further information on the radiance distribution is available. The systematic error of this approach isd = D isotr, D. The exact calculation of D requires the knowledge of the radiance distribution. For a rough estimate we assume that D is proportional to the dierence of D isotr from 1. The data of Grobner et al. [1996] can then be used to estimate D: D() j1, D isotr()j j1, D isotr;g ()j D G() (15) where D isotr;g () is and D G () is the dierence of D isotr;g () and D G () reported by Grobner et al. [1996]. Equation (15) and its underlying assumption are reasonable because D() approaches zero, when the cosine error vanishes, and approaches D G () in the case of the instrument by Grobner et al. [1996] Implementation of cosine corrections and the eect of clouds. All the above considerations refer to cloudless sky situations. To study the eect of clouds, Bernhard and Seckmeyer [1997] have carried out measurements with two almost identical spectroradiometers. One was equipped with a diuser, which had a relatively high cosine error (D isotr () = 0:903), and the second instrument used newly developed entrance optics with low cosine error (D isotr () = 0:981). For a day with cloudless sky the dierence between the uncorrected measurements of both instruments was up to 12%. This dierence was explained by the cosine error, and, consequently, correction factors K C () = 1=R C () were applied to the measurements. These were calculated with (12), where q() was derived from a model and D() was approximated by D isotr (). For a day with cloudless sky, K C (350 nm) of the instrument with the diuser varied between and 1.134, depending on the SZA. K C (350 nm) for the second instrument was between 1.00 and After the correction the results of both instruments agreed to within 2% at 350 nm. This good agreement gives condence in the way the correction factors were calculated for the particular cloudless day chosen by Bernhard and Seckmeyer [1997]. Bernhard and Seckmeyer [1997] set q() = 0 for overcast situations, and assumed that the sky radiance is isotropic. Again, the dierence in the results of both instruments can be explained. This indicates that the assumption of isotropic sky radiance is valid, although it can be expected that under homogenous clouds, zenith radiance is likely to be greater than that originating near the horizon [Grant and Heisler, 1997]. For the day with broken cloud chosen by Bernhard and Seckmeyer [1997], the ratio of the uncorrected measurements of both instruments alternates between the ratio for clear sky and overcast sky, in dependence of whether the Sun is visible or not. If no information about Sun visibility exists, we suggest setting K cloudy C = 1=2(KC clear + KC overcast ), where K cloudy C, KC clear (), and KC overcast () are the correction factors for broken cloud, clear-sky, and overcast situations (i.e., q() = 0), respectively. We propose to set the uncertainty u cloudy K C of as follows: K cloudy C u cloudy K C = u clear K C p 3 jkclear C (),K overcast C ()j; (16) where u clear K C is the uncertainty of the correction factor for cloudless sky, calculated with (13). The second term in (16) describes the additional uncertainty introduced

9 SOLAR UV MEASUREMENT UNCERTAINTY 9 by cloud variability; its formulation is based on (7). Since it cannot be expected that both terms in (16) are independent, they are added rather than combined in quadrature. As a consequence of the underlying data set, which does not allow appropriate statistics, the proposed method to estimate u cloudy K C is itself the subject of considerable uncertainties and may therefore not be appropriate for all conceiveable cloud situations Spectral Resolution The entrance and exit apertures of a monochromator have nite widths. As a consequence, not only do photons with the desired wavelength 0 pass through the monochromator but also those with wavelengths inside a certain interval around 0. When the monochromator is set to a xed wavelength 0, its transmittance as a function of wavelength is called the slit function f(). The width of the slit function is usually quantied by the full width of the function at half of its maximum (FWHM) and is denoted the bandwidth B of the monochromator. In practice, the slit function is usually determined by scanning a monochromatic radiation source (e.g., a selected line from a low-pressure Mercury lamp or a laser) by the spectroradiometer. Neglecting changes in the spectral responsivity over the scanning range, the slit function is the mirror image of the recorded spectrum with the mirror plane at the wavelength of the incident radiation. Typical bandwidths of current UV spectroradiometers are between 0.3 and 2.0 nm FWHM [Webb, 1997]. Since most Fraunhofer lines of the solar spectrum are narrower than 0.01 nm (see Figure 2), the Fraunhofer structure is smoothed out considerably by these instruments. The measured irradiance E R (; f) is the true irradiance E() convolved with the instrument's slit function f(): E R (; f) = Z E( 0 ) f(, 0 ) d 0 Z f( 0 ) d 0 (17) In the example of Figure 2 a high-resolution solar spectrum is compared with the same spectrum convolved with the slit function of the mobile IFU spectroradiometer, which has a FWHM of nm, thus mimicking a spectral measurement of the IFU instrument. The high-resolution spectrum was obtained with the Fourier transform spectrometer at the McMath/Pierce Solar Telescope situated on Kitt Peak, Arizona [Kurucz et al., 1984]. The resolution of this instrument is about 0.01 cm,1 in the UV, corresponding to nm at 350 nm. Since most Fraunhofer lines are broader than nm [Moore, 1966], the reproduction of the Fraunhofer structure is hardly inuenced by the resolution of the Kitt Peak spectrum. The ratio of both data sets in Figure 2 reaches values of up to 14. There is some controversy as to whether this Spectral irradiance [mw/(m 2 nm)] McMath/Pierce Spectrum, original resolution McMath/Pierce Spectrum, convolved Wavelength [nm] Figure 2. High-resolution solar spectrum in the region of the calcium doublet obtained with the Fourier transform spectrometer at the McMath/Pierce Solar Telescope situated on Kitt Peak, Arizona. The thin line represents the spectrum in original resolution, while the thick line denotes the spectrum convolved with the slit function of the mobile Fraunhofer Institute for Atmospheric Environmental Research (IFU) spectroradiometer of nm full width at half maximum (FWHM). deviation should be regarded as an error or whether it is better denoted as an instrumental feature: it has to be named an error if the quantity to be measured is global spectral irradiance E G () as dened in (1). Obviously, the problem disappears if the measurand is \spectral irradiance referred to the instrument's slit function." The more suitable denition of the measurand depends on the application of the data. For example, for a meaningful comparison of measured spectra with a model it is advisable to convolve the model results with the instrument's slit function before forming the ratio of both spectra [Mayer et al., 1997]. The comparision is then free from the inuence of the instrument's resolution. If measurements of instruments with dierent resolution are intercompared it may also be useful to normalize their results to a common resolution by means of deconvolution techniques [Slaper et al., 1995]. On the other hand, if the measured data are used to calculate values of biologically weighted irradiance (e.g., erythemal irradiance), the resolution is no longer negligible. For example, owing to the strong decline of the solar spectrum in the UVB, the bandwidth causes a systematic overestimate of spectral irradiance, leading to results of biologically weighted irradiance that are too high. This will be quantied in the following Errors in measurements of biologically weighted irradiance due to the resolution of the spectroradiometer. These errors are quantied with the factors R R;Bio (B), which are dened as

10 10 BERNHARD AND SECKMEYER the ratio of biologically weighted irradiance determined from measurements of a spectroradiometer with a slit function f() to the respective values from an ideal instrument with an innitesimal bandwidth: DU R R;Bio (B) = R 1 0 E R (; f) A Bio () d R 1 E() A 0 Bio () d ; (18) where A Bio () is the relevant biological action spectrum and B is the FWHM of the slit function f(). Two action spectra were considered. One is the Commission Internationale de l'eclairage (CIE) action spectrum for erythema A Ery () [McKinlay and Diey, 1987]; the second is the action spectrum of DNA damage A DNA () [Setlow, 1974], parameterized according to Bernhard et al. [1997]: A DNA () = (, i 1 h13:82 exp 1 0: exp [( [nm],310)=9],1 ;370 nm 0 ;>370 nm (19) The ratio R R;Bio (B) depends not only on bandwidth B but also on the spectral shape of the underlying solar spectrum E(). With model sensitivity studies based on the UVSPEC model we have conrmed that from all parameters determining biologically weighted irradiance at the ground, mainly solar elevation and total ozone column aect R R;Bio (B). The variation of R R;Bio (B) introduced by changes of surface albedo between 0 and 1, aerosol optical depth (320 nm) between 0 and 1.8, and altitude between 0 and 3000 m is about a factor of 15 smaller than the variation introduced by solar elevation and ozone. Also, the inuence of thin clouds is negligible. However, the attenuation of spectral irradiance by convective clouds with high optical density (e.g., thunderstorm clouds) is signicantly enhanced in the UVB relative to the UVA [Mayer et al., 1998], which leads to changes of R R;Bio (B) aswell. To quantify the inuence of solar elevation and total ozone on R R;Bio (B), model spectra were calculated for solar elevations between 0 and 90 and ozone values between 250 and 400 DU. From the modeled irradiance E(), E R (; f) was calculated for triangular slit functions with bandwidths ranging from 0.3 to 5.0 nm FWHM. Finally, the factors R R;Bio (B) were calculated for the DNA and erythema action spectra. The DNA weighted ratios R R;DNA (B) are shown in Figure 3 as a function of solar elevation and ozone column. For the slit function of bandwidth 2 nm, R R;DNA (2 nm) varies between and The respective values for the erythemal weighting range between and Thus DNA weighted irradiance can be overestimated by up to 4%, if the bandwidth of the spectroradiometer is 2 nm FWHM and no corrections are applied. The factors R R;Bio (B) were found to depend almost quadratically on bandwidth B and thus can be approx- Ratio DU R R,DNA (2.0 nm) R R,DNA (1.0 nm) R R,DNA (0.5 nm) R R,DNA (0.3 nm) R R,DNA (2.0 nm) Solar elevation [degrees] Figure 3. Eect of resolution on DNA weighted irradiance. Shown are ratios R R;DNA (B) of measured and \true" DNA weighted irradiance as a function of solar elevation and total ozone for four dierent bandwidths between 0.3 and 2.0 nm. For each solar elevation a subset of seven data points is drawn, referring to the ozone values of (left to right) 250, 280, 300, 320, 340, 370, and 400 DU. The ratios R? R;DNA (2 nm) approximate R R;DNA (2 nm) according to equation (20). imated by R? R;Bio (B): R? R;Bio(B) = 1+[R R;Bio (1 nm), 1] B 2 ; (20) where B is in nanometers. For bandwidths between 0.5 and 2.0 nm, [R R;DNA (B), 1] and [R? R;DNA (B), 1] agree to within 5%. To illustrate the good agreement, R? R;DNA (2 nm) is drawn in Figure 3 as well. In Table 2, R R;DNA (1 nm) and the analogous quantity for erythemal weighting, R R;Ery (1 nm), are given for several solar elevations and ozone values. With (20) these values can be estimated for triangular slit functions with bandwidths in between 0.5 and 2.0 nm with an accuracy of better than 5% Spectral irradiance errors caused by the resolution of the spectroradiometer. The ratio of spectral irradiance measured with a spectroradiometer with slit function f() and the \true" spectral irradiance is R R (; B) =E R (; f)=e(), where, again, B is the bandwidth of the slit function f(). Owing to the Fraunhofer ne structure, these ratios vary on a nm scale. Because of this spectral structure, Webb et al. [1998] conclude that uncertainties caused by the slit function cannot easily be dened. In order to parameterize the systematic overestimate of the true solar spectrum at short wavelengths caused by the slit function, we have tted exponential functions R R? (; B) to the ratios R R (; B). With these ts the general slope of R R (; B) at the ozone cuto and the dependence of this ratio on solar elevation and total ozone are illustrated:

11 SOLAR UV MEASUREMENT UNCERTAINTY 11 Table 2. Parameters to Calculate Errors Caused by Finite Resolution of a Spectroradiometer a and Wavelength Shifts b on Dependence of Solar Elevation (Sh) and Total Ozone (Oz) Sh Oz, DU RR;DNA(1nm) RR;Ery(1nm) cr;1 cr;2 RW;DNA(0:1nm) RW;Ery(0:1nm) cw;1 cw; Mean c Standard deviation c a See section 3.3. b See section 3.4. c Means and standard deviations of the parameters are given over the whole ranges of solar elevations (10-90 ) and ozone colums ( DU). R? R(; B) = 1+fexp[(c R;1, )=c R;2 ]g B 2 ; (21) where c R;1 and c R;2 are coecients depending on solar elevation and ozone, respectively, and and B have to be set in nanometers; c R;1 and c R;2 are given in Table 2. The denition of smooth t functions is reasonable because systematic errors in biologically weighted irradiance calculated with these functions agree well with the correct calculation using (18): for a bandwidth of 1 nm the avarage value of R R;DNA (1 nm) calculated with (18) is 1: :00077 (see Table 2). If E R (; f) in (18) is substituted by the approximation E() R R? (; 1nm) and R R;DNA(1 nm) is calculated again, the result is 1: : In Figure 4, R R? (; 1 nm) is shown for three dierent solar elevations and ozone columns between 250 and 400 DU. Generally, the ratios increase strongly when decreases: at 300 nm, 30 solar elevation, and 400 DU, R R? (; 1nm) is 1.087, whereas at 315 nm the ratio is almost 1. With increasing solar elevation or decreasing ozone the curves are shifted toward shorter wavelengths Eect of nontriangular slit functions. Up until now, only symmetrical slit functions of triangular shape were considered. Slit functions of actual spectroradiometers usually deviate from this ideal shape and are a mixture of triangular and Gaussian functions. Sensitivity studies have shown that the conclusions drawn above remain valid when the slit function is of Gaussian shape [Bernhard, 1997]. For example, at 70 solar elevation and 320 DU, R R;DNA (1 nm) is for a triangular slit function of 1 nm FWHM and for a slit function of the same bandwidth, but of Gaussian shape. However, if the slit function of an instrument appears to deviate signicantly from either shapes, the measurement errors have to be calculated explicitly with (17) and (18). In particular, instruments with an asymmerical slit function have tobe treated with special care. Here the inuence of slit function is mixed with the wavelength misalignment Wavelength Misalignment Because of the strong decline of the solar spectrum in the UVB, small errors in the wavelength alignment of a spectroradiometer lead to signicant errors in measured spectral irradiance. If a wavelength shift at a specic wavelength is well dened, the resulting systematic errors can be corrected. However, in practice, the shift is only known within certain limits (e.g., 0:05 nm), leading to uncertainties in irradiance readings. Usually, the wavelength setting of a spectroradiometer is calibrated by measuring the spectrum of a line